Textbook Question
{Use of Tech} Binomial series
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.
f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.3.21a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=3ˣ, a=0
Verified step by step guidance1
Recall that the Maclaurin series is a Taylor series centered at \(a=0\), and is given by the formula:
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n,\]
where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
Identify the function: \(f(x) = 3^x\). To find the Maclaurin series, we need to compute the derivatives of \(f(x)\) and evaluate them at \(x=0\).
Calculate the first four derivatives of \(f(x)\):
- \(f(x) = 3^x\)
- \(f'(x) = 3^x \ln(3)\)
- \(f''(x) = 3^x (\ln(3))^2\)
- \(f^{(3)}(x) = 3^x (\ln(3))^3\)
- \(f^{(4)}(x) = 3^x (\ln(3))^4\)
Evaluate each derivative at \(x=0\):
- \(f(0) = 3^0 = 1\)
- \(f'(0) = 3^0 \ln(3) = \ln(3)\)
- \(f''(0) = 3^0 (\ln(3))^2 = (\ln(3))^2\)
- \(f^{(3)}(0) = 3^0 (\ln(3))^3 = (\ln(3))^3\)
Write the first four nonzero terms of the Maclaurin series using the formula:
\[f(x) \approx f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3.\]
Substitute the values found in the previous step to express the series explicitly.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor and Maclaurin Series
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. When centered at zero, it is called a Maclaurin series. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!.
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Convergence of Taylor & Maclaurin Series
Derivatives of Exponential Functions
To find the Taylor series of f(x) = 3^x, you need to compute its derivatives at the center point. The derivative of an exponential function a^x is a^x times the natural logarithm of a. This pattern repeats for higher-order derivatives, which is essential for determining the series coefficients.
Recommended video:
04:50Derivatives of General Exponential Functions
Interval of Convergence
The interval of convergence is the set of x-values for which the Taylor series converges to the function. For exponential functions like 3^x, the series converges for all real numbers, meaning the interval of convergence is (-∞, ∞). Understanding this ensures the series accurately represents the function within this domain.
Recommended video:
08:44Interval of Convergence
Related Practice
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = tan ⁻¹ (x/2), a = 0
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function f(x) = √x has a Taylor series centered at 0.
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Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=2/(1−x)³, a=0
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Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = e⁻ˣ, a=0
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Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
a. Estimate f(0.1) and give a bound on the error in the approximation.
f(x) = eˣ ≈ 1 + x
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